What can I say to describe this game? Addictive? I think that sums it up nicely, don't you?
It starts out with you, a human character you name, on a train. The train is heading to a town (which you also will name). On the train, you meet a cat who asks a lot of questions (your answers will affect how your character looks soon). Eventually, you get the point where he finds out you don't have the means to afford a home in the town you're heading to. The cat makes a call, and gets things set up.
You arrive in the town, and meet entrepreneur Tom Nook, a raccoon. He loans you a house, as long as you pay him back. He has you do tasks for him and the townspeople, and you start earning your way to paying off the debt.
In the game, there are dozens of different people, all of varying species, who provide you with information, tasks, and even goods. You can fish, catch bugs, and even get involved in Black Market trading.
You'll find yourself immersed in finding the rare bugs and fish, the valuable items, and furniture sets. You'll spend loads of money, and also gain money in the process. It's really addicting, and if you look at it the right way, it even teaches some basic economics.
The main point of the game is to make money (Bells in the game). Seems like a terrible premise, but it's more addicting and entertaining to do. All you want to do is pay off your debt to Mr. Nook, and when you do, you are given the options of upgrading your house. You'll give yourself more debt, but to be able to store all your precious items seems worth it right?
In the end, if you fully pay off your house (including all upgrades) you get a memorial statue put up in town, honoring you for your good deeds and honesty.